chapter4
Hsiang-Leng, Wang
11/25/2019
MOOC Introduction to Open Data Science 2019: Homework4
Step1. Check the Boston dataset
View the dataset
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selecthead(Boston, n=15)## crim zn indus chas nox rm age dis rad tax ptratio black
## 1 0.00632 18.0 2.31 0 0.538 6.575 65.2 4.0900 1 296 15.3 396.90
## 2 0.02731 0.0 7.07 0 0.469 6.421 78.9 4.9671 2 242 17.8 396.90
## 3 0.02729 0.0 7.07 0 0.469 7.185 61.1 4.9671 2 242 17.8 392.83
## 4 0.03237 0.0 2.18 0 0.458 6.998 45.8 6.0622 3 222 18.7 394.63
## 5 0.06905 0.0 2.18 0 0.458 7.147 54.2 6.0622 3 222 18.7 396.90
## 6 0.02985 0.0 2.18 0 0.458 6.430 58.7 6.0622 3 222 18.7 394.12
## 7 0.08829 12.5 7.87 0 0.524 6.012 66.6 5.5605 5 311 15.2 395.60
## 8 0.14455 12.5 7.87 0 0.524 6.172 96.1 5.9505 5 311 15.2 396.90
## 9 0.21124 12.5 7.87 0 0.524 5.631 100.0 6.0821 5 311 15.2 386.63
## 10 0.17004 12.5 7.87 0 0.524 6.004 85.9 6.5921 5 311 15.2 386.71
## 11 0.22489 12.5 7.87 0 0.524 6.377 94.3 6.3467 5 311 15.2 392.52
## 12 0.11747 12.5 7.87 0 0.524 6.009 82.9 6.2267 5 311 15.2 396.90
## 13 0.09378 12.5 7.87 0 0.524 5.889 39.0 5.4509 5 311 15.2 390.50
## 14 0.62976 0.0 8.14 0 0.538 5.949 61.8 4.7075 4 307 21.0 396.90
## 15 0.63796 0.0 8.14 0 0.538 6.096 84.5 4.4619 4 307 21.0 380.02
## lstat medv
## 1 4.98 24.0
## 2 9.14 21.6
## 3 4.03 34.7
## 4 2.94 33.4
## 5 5.33 36.2
## 6 5.21 28.7
## 7 12.43 22.9
## 8 19.15 27.1
## 9 29.93 16.5
## 10 17.10 18.9
## 11 20.45 15.0
## 12 13.27 18.9
## 13 15.71 21.7
## 14 8.26 20.4
## 15 10.26 18.2View the dataset structure
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...The Boston dataset is aiming to see the housing values in suburbs of Boston which includes 14 variables and 506 rows and it contains the following columns:
- crim: per capita crime rate by town.
- zn: proportion of residential land zoned for lots over 25,000 sq.ft.
- indus: proportion of non-retail business acres per town.
- chas: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
- nox: nitrogen oxides concentration (parts per 10 million).
- rm: average number of rooms per dwelling.
- age: proportion of owner-occupied units built prior to 1940.
- dis: weighted mean of distances to five Boston employment centres.
- rad: index of accessibility to radial highways.
- tax: full-value property-tax rate per $10,000.
- ptratio: pupil-teacher ratio by town.
- black: 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town.
- lstat: lower status of the population (percent).
- medv: median value of owner-occupied homes in $1000s.
Graphical overview
Check the correlations between variables via correlation matrix plot.
## corrplot 0.84 loaded
If the relationship between two variables are more related, the circle would be more bigger and darker. According to the size and color of the circles, we can observe that those variables are related as following:
- crim: rad, tax
- zn: dis, age, indus
- indus: nox, dis, tax
- chas: N/A
- nox: age, dis, tax
- rm: latat, medv
- age: dis, lstat
- dis: rad, tax, lstat
- rad: tax
- tax: lstat
- ptratio: medv
- black: crim, rad, tax…
- lstat: rm, age…
- medv: rm
Summary Boston dataset
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Standardize the dataset
## crim zn indus chas nox rm age
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629 -0.1198948
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824 0.3668034
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456 -0.2655490
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978 -0.8090878
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620 -0.5106743
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916 -0.3508100
## dis rad tax ptratio black lstat medv
## 1 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159 -1.0744990 0.1595278
## 2 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159 -0.4919525 -0.1014239
## 3 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351 -1.2075324 1.3229375
## 4 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514 -1.3601708 1.1815886
## 5 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159 -1.0254866 1.4860323
## 6 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651 -1.0422909 0.6705582## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865Create a categorical variable of the crime rate in the Boston dataset (from the scaled crime rate)
p.s. Use the quantiles as the break points in the categorical variable
# class of the boston_scaled object
class(Boston_scale)## [1] "matrix"Boston_scale <- as.data.frame(Boston_scale)
# create a quantile vector of crim and print it
bins <- quantile(Boston_scale$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610# create a categorical variable 'crime'
crime <- cut(Boston_scale$crim, breaks = bins, include.lowest = TRUE)
table(crime)## crime
## [-0.419,-0.411] (-0.411,-0.39] (-0.39,0.00739] (0.00739,9.92]
## 127 126 126 127# remove original crim from the dataset
Boston_scale <- dplyr::select(Boston_scale, -crim)
# add the new categorical value to scaled data
Boston_scale <- data.frame(Boston_scale, crime)Divide the dataset to train and test sets, so that 80% of the data belongs to the train set
# number of rows in the Boston dataset
n <- nrow(Boston_scale)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- Boston_scale[ind,]
# create test set
test <- Boston_scale[-ind,]
# save the correct classes from test data
correct_classes <- Boston_scale[-ind,]$crime
# remove the crime variable from test data
test <- dplyr::select(test, -crime)Fit the linear discriminant analysis on the train set
- Use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables. Draw the LDA (bi)plot.
- Save the crime categories from the test set and then remove the categorical crime variable from the test dataset
# linear discriminant analysis
lda.fit <- lda(crime~., data = train)
# print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## [-0.419,-0.411] (-0.411,-0.39] (-0.39,0.00739] (0.00739,9.92]
## 0.2425743 0.2549505 0.2524752 0.2500000
##
## Group means:
## zn indus chas nox rm
## [-0.419,-0.411] 1.0674935 -0.9103987 -0.151805591 -0.8992079 0.47453529
## (-0.411,-0.39] -0.1575441 -0.2841545 -0.004759149 -0.5457371 -0.11899121
## (-0.39,0.00739] -0.3762641 0.2093424 0.190859195 0.4080625 0.02825978
## (0.00739,9.92] -0.4872402 1.0171306 -0.038441925 1.0682825 -0.45716027
## age dis rad tax ptratio
## [-0.419,-0.411] -0.9117853 0.8854215 -0.6783478 -0.7309745 -0.4701174
## (-0.411,-0.39] -0.2904168 0.2781807 -0.5514748 -0.4603536 -0.0588367
## (-0.39,0.00739] 0.4439145 -0.3830590 -0.4188972 -0.3025568 -0.2389429
## (0.00739,9.92] 0.7864958 -0.8429522 1.6379981 1.5139626 0.7806252
## black lstat medv
## [-0.419,-0.411] 0.3771190 -0.7840402 0.55317432
## (-0.411,-0.39] 0.3181237 -0.1165358 0.01195073
## (-0.39,0.00739] 0.1371315 0.1133702 0.09876697
## (0.00739,9.92] -0.7590438 0.8550059 -0.66660397
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.08507971 0.7976884508 -0.95082889
## indus 0.06688034 -0.2510151468 0.01341613
## chas -0.09512906 -0.0658670907 0.06959093
## nox 0.35492505 -0.6647559943 -1.45315646
## rm -0.08841471 -0.0823398130 -0.16216564
## age 0.25122380 -0.2788318528 -0.12868828
## dis -0.07459395 -0.3249248993 0.06019282
## rad 3.21188226 1.0038033960 -0.11482108
## tax -0.04078396 -0.0977186134 0.83568167
## ptratio 0.10629227 -0.0009783461 -0.27079419
## black -0.13180164 0.0009285333 0.08517103
## lstat 0.21620915 -0.3110895387 0.30887030
## medv 0.18169442 -0.3818443691 -0.15836034
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9433 0.0426 0.0141# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results
plot(lda.fit, dimen = 2)
lda.arrows(lda.fit, myscale = 1)
Predict the classes with the LDA model on the test data. Cross tabulate the results with the crime categories from the test set. Comment on the results.
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct [-0.419,-0.411] (-0.411,-0.39] (-0.39,0.00739] (0.00739,9.92]
## [-0.419,-0.411] 14 13 2 0
## (-0.411,-0.39] 5 16 2 0
## (-0.39,0.00739] 0 9 14 1
## (0.00739,9.92] 0 0 0 26Reload the Boston dataset and standardize the dataset
- Calculate the distances between the observations. Run k-means algorithm on the dataset. Investigate what is the optimal number of clusters and run the algorithm again.
- Visualize the clusters. For example with the pairs() or ggpairs() functions, where the clusters are separated with colors and interpret the results
library(MASS)
data('Boston')
# scale the dataset
bs <- scale(Boston)
bs <- as.data.frame(bs)
# distance measure 1: euclidean distance matrix
dist_eu <- dist(bs, method = "euclidean")
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970# distance measure 2: manhattan distance matrix
dist_man <- dist(bs, method = "manhattan")
# look at the summary of the distances
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618# k-means clustering
# plot the Boston dataset with clusters
km <-kmeans(bs, centers = 4)
pairs(bs[6:10], col = km$cluster)
# k-means clustering
km <-kmeans(bs, centers = 3)
pairs(bs[6:10], col = km$cluster)
# k-means clustering
km <-kmeans(bs, centers = 2)
pairs(bs[6:10], col = km$cluster)
# k-means clustering
km <-kmeans(bs, centers = 1)
pairs(bs[6:10], col = km$cluster)
As shown in the k-means graph above (from center=4 to 1), the pair graph with center=2 seems the most resonable seperation. In the graph when center=3, some cluster are ambiguous winthin one group.
Bonus
- Perform k-means on the original Boston data with some reasonable number of clusters (> 2).
- Perform LDA using the clusters as target classes. Include all the variables in the Boston data in the LDA model.
Boston_scale_original <- scale(Boston)
Boston_scale_original <- as.data.frame(Boston_scale_original)
# k-means clustering
km_original <-kmeans(Boston_scale_original, centers = 4)
pairs(Boston_scale_original[6:10], col = km_original$cluster)
# k-means clustering
km_original <-kmeans(Boston_scale_original, centers = 3)
pairs(Boston_scale_original[6:10], col = km_original$cluster)
Visualize the results with a biplot (include arrows representing the relationships of the original variables to the LDA solution).
Interpret the results. Which variables are the most influencial linear separators for the clusters? (0-2 points to compensate any loss of points from the above exercises)
Super-Bonus
Run the code below for the (scaled) train data that you used to fit the LDA. The code creates a matrix product, which is a projection of the data points.